High-fidelity simulations of the flow aroundwings at high Reynolds numbers

R. Vinuesa, P. S. Negi, A. Hanifi, D. S. Henningson and P.Schlatter

Department of Mechanics, Linne FLOW Centre and Swedish e-Science ResearchCentre (SeRC), KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden

In Proc. 10th Int. Sym. on Turbulence & Shear Flow Phenomenon (TSFP-10)

Reynolds-number effects in the adverse-pressure-gradient (APG) turbulentboundary layer (TBL) developing on the suction side of a NACA4412 wingsection are assessed in the present work. To this end, we conducted a well-resolved large-eddy simulation of the turbulent flow around the NACA4412airfoil at a Reynolds number based on freestream velocity and chord lengthof Rec = 1, 000, 000, with 5◦ angle of attack. The results of this simulationare used, together with the direct numerical simulation by Hosseini et al. (Int.J. Heat Fluid Flow 61, 2016) of the same wing section at Rec = 400, 000, tocharacterize the effect of Reynolds number on APG TBLs subjected to thesame pressure-gradient distribution (defined by the Caluser pressure-gradientparameter β). Our results indicate that the increase in inner-scaled edge velocityU+e , and the decrease in shape factor H, is lower in the APG on the wing than

in zero-pressure-gradient (ZPG) TBLs over the same Reynolds-number range.This indicates that the lower-Re boundary layer is more sensitive to the effect ofthe APG, a conclusion that is supported by the larger values in the outer regionof the tangential velocity fluctuation profile in the Rec = 400, 000 wing. Futureextensions of the present work will be aimed at studying the differences inthe outer-region energizing mechanisms due to APGs and increasing Reynoldsnumber.

Key words: adverse pressure-gradient, boundary-layer, wings

1. Introduction

Turbulent boundary layers (TBLs) subjected to streamwise pressure gradients(PGs) are relevant to a wide range of industrial applications from diffusersto turbines and wings, and pose a number of open questions regarding theirstructure and underlying dynamics. A number of studies over the years haveaimed at shedding some light on these open questions from the theoretical(Townsend 1956; Mellor & Gibson 1966), experimental (Skare & Krogstad 1994;Harun et al. 2013) and numerical (Spalart & Watmuff 1993; Skote et al. 1998)

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perspectives, but the large number of parameters influencing the structure ofPG TBLs raises serious difficulties when comparing databases from differentexperimental or numerical databases (Monty et al. 2011). The current work isfocused on the analysis of adverse-pressure-gradient (APG) effects on TBLs,a flow case that can be observed, for instance, on the suction side of wings.As the boundary layer develops, it encounters a progressively larger resistancemanifested in the increased pressure in the streamwise direction. This APGdecelerates the boundary layer, increases its wall-normal momentum, andincreases its thickness while reducing the wall-shear stress. As a result of thelarger boundary-layer thickness the wake parameter in the mean velocity profileincreases (Vinuesa et al. 2014), and more energetic turbulent structures developin the outer region (Maciel et al. 2017). The recent work by Bobke et al. (2017)highlights the importance of the flow development in the establishment of anAPG TBL, and in particular the streamwise evolution of the Clauser pressure-gradient parameter β = δ∗/τwdPe/dx (where δ∗ is the displacement thickness,τw the wall-shear stress and dPe/dx is the streamwise pressure gradient). In theirstudy, Bobke et al. (2017) compared different APG TBLs subjected to variousβ(x) distributions, including several flat-plate cases and one APG developing onthe suction side of a wing section (Hosseini et al. 2016). Their main conclusionwas the fact that the effect of APGs was more prominent in the cases wherethe boundary layer had been subjected to a stronger pressure gradient for alonger streamwise distance, a conclusion that demonstrates the relevance ofaccounting for the β(x) distribution when assessing pressure-gradient effectson TBLs. Along these lines, the numerical studies by Kitsios et al. (2016), Lee(2017) and Bobke et al. (2017) aim at characterizing the effect of APGs onTBLs in cases with a constant pressure-gradient magnitude, i.e., in flat-plateboundary layers exhibiting long regions with constant values of β. The aimof the present work is to assess the effect of the Reynolds number (Re) ontwo APG TBLs subjected to the same β(x). In particular, we consider theturbulent flow around a NACA4412 wing section at two Reynolds numbersbased on freestream velocity U∞ and chord length c, namely Rec = 400, 000and 1, 000, 000. As discussed below, the β(x) distribution is almost identicalin the two cases, a fact that allows to characterize the impact of Re on theboundary-layer development. The former database is a the direct numericalsimulation (DNS) by Hosseini et al. (2016), whereas the latter is a well-resolvedlarge-eddy simulation (LES) conducted in the current study, and described inthe next section.

2. Computational setup

A well-resolved LES of the flow around a NACA4412 wing section was carriedout using the spectral-element code Nek5000 (Fischer et al. 2008), developedat Argonne National Laboratory. In the spectral-element method (SEM) thecomputational domain is decomposed into elements, where the velocity andpressure fields are expressed in terms of high-order Lagrange interpolants ofLegendre polynomials, at the Gauss–Lobatto–Legendre (GLL) quadrature points.

TBL around wing sections 107

In the present work we used the PN − PN−2 formulation, which implies thatthe velocity and pressure fields are expressed in terms of polynomials of orderN and N − 2, respectively. The time discretization is based on an explicitthird-order extrapolation for the nonlinear terms, and an implicit third-orderbackward differentiation for the viscous ones. The code is written in Fortran 77and C and the message-passing-interface (MPI) is used for parallelism. We haveused Nek5000 to simulate wall-bounded turbulent flows in moderately complexgeometries in a wide range of internal (Marin et al. 2016) and external (Vinuesaet al. 2015) configurations.

A two-dimensional slice of the computational domain is shown in Figure 1(left), where x, y and z denote the horizontal, vertical and spanwise directions,respectively. The domain is periodic in the spanwise direction, with a width ofLz = 0.2c. A total of 4.5 million spectral elements was used to discretize thedomain with a polynomial order N = 7, which amounts to a total of 2.3 billiongrid points. As in the DNS simulation by Hosseini et al. (2016), a Dirichletboundary condition extracted from an initial RANS (Reynolds-Averaged Navier–Stokes) simulation was imposed on all the boundaries except the outflow, wherethe boundary condition by Dong et al. (2014) was employed. The initial RANSsimulation was carried out with the k − ω SST (shear-stress transport) model(Menter 1994) implemented in the commercial software ANSYS Fluent. In thecurrent configuration, a Reynolds number of Rec = 1, 000, 000 was considered,together with an angle of attack of 5◦. The LES approached is based on arelaxation-term (RT) filter, which provides an additional dissipative force inorder to account for the contribution of the smallest, unresolved, turbulent scales(Schlatter et al. 2004). A validation of the method in turbulent channel flowsand the flow around a NACA4412 wing section is given by Negi et al. (2017).The mesh resolution around the wing follows these guidelines: Δx+ < 27,Δy+wall < 0.96 and Δz+ < 13, where the superscript ‘+’ denotes scaling in terms

of the friction velocity uτ =�τw/ρ (with ρ being the fluid density). Regarding

the wake, we defined the criterion Δx/η < 13, where η =�ν3/ε

�1/4is the

Kolmogorov scale (ν is the fluid kinematic viscosity, and ε the local isotropicdissipation). An instantaneous flow field showing the coherent structuresidentified with the λ2 method (Jeong & Hussain 1995) is shown in Figure 1(right), which also highlights the adequacy of the present LES approach tosimulate this flow. Note that the boundary layers on the suction and pressuresides were tripped using the volume-force method described by Schlatter &Orlu (2012).

3. Results and discussion

As discussed in the introduction, the aim of the current study is to investigate theReynolds-number effects on APG TBLs subjected to the same β(x) distribution.In particular, we aim at assessing such effects on the turbulent boundary layerdeveloping on the suction side (denoted as ss) of a NACA4412 wing section with5◦ angle of attack. To this end, we compare the results from the DNS database

108 Vinuesa et al.

Figure 1: (Left) Two-dimensional slice of the computational domain showingthe spectral-element distribution, but not the individual GLL points. (Right)Instantaneous flow field showing coherent structures identified with the λ2

method (Jeong & Hussain 1995), and colored with horizontal velocity. In thisfigure, dark blue represents a horizontal velocity of −0.1 and dark red a valueof 2.

at Rec = 400, 000 by Hosseini et al. (2016) with the current well-resolved LESat Rec = 1, 000, 000. The turbulence statistics presented in this study for theRec = 1, 000, 000 case were obtained after averaging for one flow-over time(where the time is non-nondimensionalized in terms of U∞ and c). Note that thespanwise width of the current simulation is twice as large as the one consideredby Hosseini et al. (2016), a fact that effectively increases the statistical samplesby a factor of two. Although this averaging period does not allow to obtainconverged turbulent kinetic energy (TKE) budgets, the mean and fluctuatingprofiles discussed here start to exhibit convergence up to xss/c � 0.7. Theboundary-layer development, mean velocity and Reynolds-stress profiles arediscussed in the next sections.

3.1. Boundary-layer development

In Figure 2 (top left) we show the streamwise evolution of the Clauer pressure-gradient parameter β for the TBLs on the suction side of the two wing casesunder study. As expected, the two boundary layers are subjected to almostidentical β(x) distributions, with small relative differences only arising beyondxss/c > 0.9. Note that the boundary layers are subjected to conditions close tozero pressure gradient (ZPG) up to xss/c � 0.3, point after which the value ofβ increases beyond 0.1. In the next section we will study the velocity profilesat xss/c = 0.4 and 0.7, in which the pressure-gradient magnitude is moderate(β � 0.6) and strong (β � 2), respectively. Although the value of β increasesthroughout the whole suction side of the wing, an inflection point is observed atxss/c = 0.4, which is the point of maximum camber in the NACA4412 airfoil.Beyond this point, the rate of change of β increases significantly with x, afact that is explained by the progressive reduction in airfoil thickness, whichproduces a larger increase in streamwise adverse pressure gradient.

TBL around wing sections 109

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Figure 2: Streamwise evolution of (top left) the Clauser pressure-gradientparameter β, (top right) the Reynolds number based on momentum thicknessReθ and (bottom) the friction Reynolds number Reτ , for the two wing casesunder study.

In Figure 2 (top right) and (bottom) we show the streamwise evolution of theReynolds number based on momentum thickness Reθ, and the friction Reynoldsnumber Reτ = δ99uτ/ν, respectively. Note that δ99 is the 99% boundary-layerthickness, which was determined following the method described by Vinuesaet al. (2016) for pressure-gradient TBLs. The Reθ trends increase monotonicallyin the two boundary layers, due to the fact that both Reynolds number andAPG promote the increase of the boundary-layer thickness. In particular, thethickenning experienced by the TBLs due to the APG significantly increasesReθ in both cases, up to a maximum value of 2, 800 in the Rec = 400, 000 case,and up to Reθ = 6, 000 in the higher-Rec wing, both observed close to thetrailing edge. Regarding the friction Reynolds number, note that in the twoboundary-layer cases the maximum is located at xss/c � 0.8, and not close tothe trailing edge as in Reθ. This is due to the fact that, although the APGincreases the boundary-layer thickness, it also decreases the wall-shear stress;thus, the very strong APGs beyond xss/c � 0.8 (where β � 4.1 in both cases)produce a larger reduction in uτ than the increase in δ99. The maximum Reτvalues are 373 and 707 in the Rec = 400, 000 and 1, 000, 000 wings, respectively.

The skin-friction coefficient Cf = 2 (uτ/Ue)2(where Ue is the velocity at the

boundary-layer edge) and the shape factor H = δ∗/θ are shown, as a functionof the streamwise position on the suction side of the wing, in Figure 3. The

110 Vinuesa et al.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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Figure 3: Streamwise evolution of (left) the skin-friction coefficient Cf and(right) the shape factor H, for the two wing cases under study.

Cf curves show different trends up to xss/c � 0.2, a fact that is explainedby the volume-force tripping at xss/c = 0.1. In the present high-Re case, the

tripping parameters were chosen following the work by Schlatter & Orlu (2012)in ZPG TBLs; however, in the Rec = 400, 000 wing the number of modes inthe spanwise direction was larger than in Schlatter & Orlu (2012), a fact thatproduces a long intermittent region in the post-transitional region. Nevertheless,the boundary layers can be considered to be essentially independent of thetripping beyond xss/c � 0.2 (Vinuesa et al. 2016). It can be observed thatthe Cf curve in the high-Re wing is below the one of the lower-Re case upto xss/c � 0.9, point after which the differences between both curves aresignificantly reduced. Since the two boundary layers are subjected to the sameβ(x) distribution, it can be argued that the differences between both curves aredue to Reynolds-number effects, a fact that is consistent with what is observedin ZPG TBLs since Cf decreases with Re. Interestingly, the effect of Reynoldsnumber becomes essentially negligible beyond xss/c � 0.9, where the very strongAPG conditions (with a value of β � 14 at xss/c = 0.9) define the state ofthe boundary layer. Regarding the shape factor, note that APG and Reynoldsnumber have opposite effects on a TBL: whereas the former increases H (dueto the thickening of the boundary layer caused by the increased wall-normalmomentum), the latter decreases the shape factor. This can also be observedin Figure 3 (right), where the H curve from the Rec = 1, 000, 000 is below theone from the Rec = 400, 000 throughout the whole suction side of the wing.Note that, since the two boundary layers are subjected to essentially the samepressure-gradient effects, the lower values of H are produced by the higherReynolds number. The differences between the effects of β and Re on TBLsare further discussed in the next section.

3.2. Inner-scaled mean velocity and Reynolds-stress profiles

Figure 4 shows the inner-scaled mean velocity profiles at xss/c = 0.4 and 0.7for the wing cases, where in the former the value of β is around 0.6 and in thelatter β � 2. Note that U+

t is the inner-scaled mean velocity in the directiontangential to the wing surface, whereas y+n is the inner-scaled wall-normal

TBL around wing sections 111

coordinate. In Figure 4 (left) we show the two wing profiles, with Reτ = 242and 449, together with ZPG TBL profiles at matched Reτ obtained from theDNS database by Schlatter & Orlu (2010). These comparisons are aimed atassessing the effect of the APG with respect to the baseline ZPG case, andalthough this comparison can be done by matching several quantities (such asReδ∗ or Reθ), in the present work we fixed Reτ as in the studies by Monty et al.(2011), Harun et al. (2013) or Bobke et al. (2017). Note that by fixing Reτwe compare two boundary layers which essentially exhibit the same range ofspatial scales, but subjected to different pressure-gradient conditions. The firstnoticeable conclusion is the more prominent wakes present in the APG TBLscompared with the corresponding ZPG TBLs at the same Reτ , which is due tothe lower skin-friction coefficient caused by the boundary-layer thickenning dueto the APG. A first step towards characterizing the effect of Re in the TBLssubjected to this particular β(x) distribution is to observe the evolution of U+

e

between Reτ = 242 and 449 in the ZPG and in the APG cases: in the former,the increase in the inner-scaled edge velocity is 11%, whereas in the latter itis 9.7%. On the other hand, the decrease in H is 3.1% in the ZPG boundarylayers, whereas the APG cases experience a larger decrease in shape factor of5.9%. These observations are also present in the profiles at xss/c = 0.7 shownin Figure 4 (right), where the Reτ values are 356 and 671 in the Rec = 400, 000and 1, 000, 000 wing cases, respectively. At this location, the increase in U+

e isaround 9.7% in the ZPG boundary layer, whereas in the APG case this increaseis 8.8%. Moreover, the shape factor decreases by 2.5% from Reτ = 356 to 671in the ZPG boundary layer, whereas the wings exhibit a larger decrease of 4.5%.On the one hand, the shape factor is larger in APG TBLs, and decreases withReynolds number as in ZPGs (which are PG TBLs with β = 0). Interestingly,the decrease in the APG case is more pronounced than the one observed inZPG boundary layers, a fact that suggests that the values of H in the low-Reboundary layer are more severely affected by the APG than the ones at higherReynolds numbers. On the other hand, the values of the inner-scaled edgevelocity increase both with Reynolds number and with the APG, since in bothcases the boundary layer grows and experiences a reduction in the velocitygradient at the wall. The fact that the increase in U+

e is larger in the ZPG casethan in the APG indicates that in the low-Reynolds-number case the boundarylayer experienced a stronger effect of the pressure gradient, therefore exhibitinga larger value of U+

e which led to a lower increase than in the β = 0 case.Thus, the evolution of U+

e and H indicates that the low-Re boundary layer ismore sensitive to the effect of the pressure gradient than the high-Re, whenboth boundary layers were subjected to the same β(x) distribution. Additionalsupport for this claim can be found in the mean velocity profiles at y+n � 25,where the ZPG cases and the high-Re wing exhibit almost identical values ofthe inner-scaled velocity U+

t , but the low-Re wing shows values below these inthe two streamwise positions. Lower velocities in the buffer layer with respectto the ZPG are associated with strong effects of the APG, as documented forinstance by Spalart & Watmuff (1993) or Bobke et al. (2017). Since these lower

112 Vinuesa et al.

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Figure 4: Inner-scaled mean velocity profiles at (left) xss/c = 0.4 and (right)xss/c = 0.7 for the two wing cases under study, compared with the DNS

results of ZPG TBL by Schlatter & Orlu (2010) at matched Reτ values.

velocities are significant in the Rec = 400, 000 wing, it can be stated that theeffect of the APG is more pronounced in this case than in the high-Re wing.

As discussed by Harun et al. (2013) or Bobke et al. (2017), the APGenergizes the outer region of the boundary layer, producing more energeticturbulent structures. This effect is also observed when increasing the Reynoldsnumber in a ZPG TBL, since as the boundary layer develops the outer regionexhibits more energetic structures as shown for instance in the experiments byHutchins & Marusic (2007) and the numerical simulations by Eitel-Amor et al.(2014). However, the mean velocity profiles shown in Figure 4 suggest thatthere may be differences in the way that this energizing process takes place,since the the evolution of the mean flow parameters with Reynolds numberis not the same in the β = 0 (ZPG) as in the APG cases. In particular, it isinteresting to note that at low Reynolds numbers the effect of the APG appearsto be more prominent than at higher Re. Large-scale energetic motions developin ZPG TBLs at increasing Reynolds number together with the developmentof the outer region of the boundary layer. The present results suggest thatsuch development of the outer region takes place in a different way when anAPG is present, a fact that is closely connected to the much larger wall-normalconvection in APGs. In APG TBLs there are two complementing mechanismsresponsible for the development of the boundary-layer outer region, namely dueto β and due to Re. In order to further analyze the differences between thesemechanisms, several components of the Reynolds-stress tensor are shown forthe two wing cases at xss/c = 0.4 and 0.7 in Figure 5. Note that we also showthe inner-scaled streamwise velocity fluctuation profiles from the ZPG DNSby Schlatter & Orlu (2010) at matched Reτ values for comparison. The firstimportant conclusion that can be drawn from Figure 5 is the fact that all thecomponents of the Reynolds-stress tensor exhibit a more energetic outer regionin comparison with ZPG TBLs, as discussed for instance by Kitsios et al. (2016)

TBL around wing sections 113

or Bobke et al. (2017). Moreover, in Figure 5 (left) it can be observed that the

increase in the near-wall peak of the tangential velocity fluctuation profile u2t

+

from Reτ = 242 to 449 is of around 4.5%, which interestingly is approximatelythe same increase as in the wing cases. In fact, and as discussed by Eitel-Amoret al. (2014), the wall-resolved LES method employed in the present study

slightly attenuates the near-wall peak of u2t

+, a fact that would indicate that

the increase in the wing boundary layers is slightly larger than in the ZPG.On the other hand, the Rec = 400, 000 wing exhibits a much more energeticouter region than the corresponding ZPG case at the same Reτ : for instance,

at y+n = 100 the low-Re wing case shows a u2t

+value 41% larger than the ZPG

at the same location. On the other hand, this difference is significantly lower

in the high-Re wing, where the u2t

+is only around 17% higher than the ZPG

at y+n = 200 (note that this wall-normal location corresponds to yn/δ99 � 0.29,i.e., approximately the same distance from the wall in outer units as in thelow-Re case). This suggests that in the low-Re APG there is a higher energyconcentration in the outer region than in the high-Re one. This is furtherconfirmed by the results shown in Figure 5 (right) at xss/c = 0.7, where the

Reτ values are 356 and 671. Firstly, the increase in the near-wall peak of u2t

+

is slightly larger in the APG boundary layers (5.1%) than in the ZPG (4.5%),a difference that could be larger if the fact that the well-resolved LES slightlyattenuates the near-wall peak in the high-Re case. However, the most significantresult in Figure 5 (right) is the fact that both APG boundary layers exhibita plateau in the outer region of the tangential velocity fluctuation profile. In

particular, the u2t

+value in this plateau is larger in the lower-Re wing (5.75)

than in the high-Re case (5.0). Since the high-Re wing exhibits a larger valueof the near-wall peak, the ratio between this maximum and the plateau in theouter region is significantly larger in the Rec = 1, 000, 000 case (1.78) than inthe Rec = 400, 00 wing (1.48). This is a very relevant result, since it shows notonly that the energizing mechanisms of the outer region in the boundary layerare different when they are connected to APG than when they are associatedto Re, but also that lower-Re TBLs are more sensitive to pressure-gradienteffects than high-Re ones. In particular, the tangential velocity fluctuationprofiles show larger values in the outer region in the lower-Re case, which is amanifestation of more prominent energy accumulation in the large-scale motionsthan in the high-Re APG boundary layer.

4. Summary and conclusions

The present study is aimed at further understanding the mechanisms responsiblefor the development of the outer region of TBLs and for the energizing of thelarge-scale motions, as well as their connection with APGs and increasingReynolds number. To this end, we performed a well-resolved LES of the flowaround a NACA4412 wing section at Rec = 1, 000, 000, with 5◦ angle of attack,using the spectral-element code Nek5000. The setup is similar to the one

114 Vinuesa et al.

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Figure 5: Selected components of the inner-scaled Reynolds-stress tensor at(left) xss/c = 0.4 and (right) xss/c = 0.7 for the two wing cases under study,

compared with the DNS results of ZPG TBL by Schlatter & Orlu (2010)at matched Reτ values. The Reynolds stresses are represented as:tangential wall-normal and spanwise velocity fluctuations, and

Reynolds shear stress.

employed by Hosseini et al. (2016) to perform a DNS of the same flow caseat a lower Rec = 400, 000. The boundary layers developing on the suctionside of the two wing sections are subjected to essentially the same streamwiseClauser pressure-gradient distribution β(x), a fact that allows to characterizethe effect of the Reynolds number in APG TBLs subjected to an increasingAPG magnitude.

As a TBL develops, the increasing Reynolds number produces a moreenergetic outer region, a fact that is manifested in the Reynolds-stress tensorprofiles. On the other hand, an APG also produces more energetic large-scalemotions in the outer region of the boundary layer due to the lift-up effect andthe increased wall-normal convection associated to it. Our results indicate thatthe skin-friction curve from the wing at Rec = 1, 000, 000 is below the one atRec = 400, 000 (up to around xss/c � 0.9), a fact that is consistent with thewell-known effect of Reynolds number in ZPG TBLs. Moreover, the shapefactor curve in the high-Re wing is also below the one at Rec = 400, 000, whichis associated with another effect of Reynolds number, i.e., to reduce H.

We also analyzed the inner-scaled mean velocity profiles at xss/c = 0.4 and0.7, which are subjected to β values of 0.6 and 2, respectively. At xss/c = 0.4,the increase of U+

e from Reτ = 242 to 449 is 9.7%, which is lower than theincrease in ZPG TBLs over the same Reτ range (11%). Similarly, at xss/c = 0.7the increase in U+

e from Reτ = 356 to 671 is 8.8%, also below the one in ZPGs,which is 9.7%. On the other hand, the shape factor is reduced at xss/c = 0.4by 5.9% and at xss/c = 0.7 by 4.5% (compared to only 3.1% and 2.5% in thecorresponding ZPG case). The steeper decrease in H and the more moderateincrease in U+

e compared to ZPG TBLs indicate that the lower-Re APG is more

TBL around wing sections 115

sensitive to pressure-gradient effects than the high-Re one. This conclusion issupported by the observations on several components of the Reynolds-stresstensor, in particular in the tangential velocity fluctuation profile. Our results

show that at xss/c = 0.4 the lower-Re wing exhibits a larger ratio of u2t

+in the

outer region with respect to the corresponding ZPG case than the high-Re case,again indicating a more pronounced effect of the APG on the lower Reynoldsnumber wing. Regarding the profiles at xss/c = 0.7, it is interesting to note

that although the high-Re wing exhibits a larger near-wall peak in u2t

+than

the lower-Re case, the latter exhibits larger values in the outer region. Thus,while the former shows a value of 5.0 in the plateau located in the outer part

of the u2t

+profile, the latter exhibits a higher value of 5.75. Consequently, the

ratio between the near-wall peak and the plateau in the u2t

+profile is 1.78 in

the high-Re wing, and 1.48 in the lower-Re case. This shows that the energydistribution in the two wing boundary layers, subjected to the same β(x), issignificantly different. Further analysis of these results will help to elucidatethe differences in the mechanisms for outer-region energizing due to APG andReynolds number.

Acknowledgments

The simulations were performed on resources provided by the Swedish NationalInfrastructure for Computing (SNIC) at the Center for Parallel Computers(PDC), in KTH, Stockholm. RV and PS acknowledge the funding provided bythe Swedish Research Council (VR) and from the Knut and Alice WallenbergFoundation. This research is also supported by the ERC Grant No. “2015-AdG-694452, TRANSEP” to DH.

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Appendix A. Domain Validation

A.1. Numerical setup

Validation study is performed in order to study the influence of domain trunca-tion on the LES of flow around sections. The numerical setup is described indetail earlier in section 2 and the setup for the domain validation simulationsfollows a very similar approach. The airfoil used for the study is ED36F128designed at the Aeronautics and Vehicle Engineering department at KTH, Stock-holm, where several experiments have been performed for both static (Lokatt &Eller 2017) and dynamic flow cases (Lokatt 2017). The chord-based Reynoldsnumber used in the investigation is Rec = 100, 000 and the angle of attack isset to α = 6.7◦. The spanwise width of the domain for all simulations was setto lz/c = 0.1. The flow is tripped to induce flow transition on both the suctionand pressure side of the airfoil at a chord-wise location of x/c = 0.1. Data froma RANS simulations with the k-ω SST turbulence model is used to obtain thevelocity field and the kolmogorov length scale η. The RANS velocity field datais imposed as Dirichlet boundary condition on the inlet and far-filed boundaries.Periodic boundary conditions are applied on the spanwise boundaries of thedomain. Simulations are performed using Nek5000 (Fischer et al. 2008) with an11th order polynomial discretization. Velocity and length scales are normalizedby free-stream velocity and chord length respectively. The grid resolution inthe boundary layer around the airfoil is set such that Δx+ ≈ 18, Δy+w ≈ 0.6,Δz+ ≈ 9. In order to account for the varying shear-stress values over the airfoilthe following rules are used for the determination of the resolution:

1. The uτ values from the RANS are used without modification in theregion 0.1 ≤ x/c ≤ 0.6 for both the pressure (ps) and suction sides (ss).

2. For x/c > 0.6, the uτ from the pressure-side was used to design the meshon both the suction and the pressure sides. The strong adverse-pressuregradient on the suction-side causes flow separation and uτ ≈ 0. Thusthe pressure-side is used to design the mesh on both sides.

3. For x/c < 0.1, a constant uτ value was used which was equal to thevalue at x/c = 0.1 for both the suction and pressure sides.

4. The spectral-element distribution is uniform in the span-wise directionwith the value of uτ at x/c = 0.25 used for calculating the Δz+ spacing.

5. In the wake the criteria is based on the kolmogorov length scale η. Theresolution in the wake is set such that Δx,Δy,Δz ≤ 10η. The value ofη is estimated from the RANS solution.

A.2. Validation cases

Table 1 lists the different domain sizes tested in the present investigation alongwith the boundary condition used at the outlet boundary of the domain. Alldistances are measured from the leading-edge. All computational domainscontain a curved inflow due to the C-grid type mesh topology. Thus the inletdistance is a radial distance from the leading-edge. A large reference case is set

TBL around wing sections 119

up in order to compare the results of the simulations with domain truncation.The stress-free boundary condition is used at the outlet for the reference case.In the subsequent test cases the energy-stabilizied outflow boundary conditionas suggested by Dong et al. (2014) is used. The boundary condition has beenshown to improve accuracy and numerical stability in severely truncated domains(Dong et al. 2014). A 2-D cross-section of the grid for the reference domain is

No. Case Name Inlet Far-field Outlet Outlet B.C.

1 Reference 5c 5c 5c Stress-free2 FF1W3 1c 1c 3c Dong3 FF2W4 2c 2c 4c Dong

Table 1: Validation cases

visualized in figure 6. Iso-contours of the instantaneous vortical structures inthe flow-field for the reference case are depicted in figure 7. All simulations arerun for 20 simulation time units and all statistics from the first 10 time unitsare discarded to remove initial transient effects.

(a) (b)

Figure 6: The simulation domain (a) and a close-up (b) near the leading edge,of the orthogonal and structured spectral-element grid for the reference case.

In the case FF1W3 the domain is truncated such that the far-field andinlet boundaries are one chord away from the airfoil (FF = 1) and the wakeregion is truncated to three chords downstream of the leading-edge (W = 3).Comparison of the wall-shear stress with the Reference case is shown in figure 8.Even with such a severely truncated computational domain, it is found thatthe flow field is only marginally disturbed with respect to the Reference case.The wall shear-stress (τw) deviates only slightly from the reference values andthis deviation occurs near a local peak of τw at x/c ≈ 0.24.

For the case FF2W4 the domain size is slightly increased such that thefar-field is now two chords away (FF = 2) and the wake region is truncated

120 Vinuesa et al.

Figure 7: Visualization of the instantaneous flow structures identified bythe λ2 criterion Jeong & Hussain (1995). The isocontours are colored bystreamwise velocity.

(a)

0 0.2 0.4 0.6 0.8 1-2

0

2

4

6

8

1010

-3

Reference

FF=1,W=3

(b)

0.2 0.25 0.35

5.5

6

6.5

7

7.510

-3

Reference

FF=1,W=3

Figure 8: The suction-side wall shear-stress comparison for the case FF1W3depicting (a)the chord-wise variation of τw and (b) the close-up region nearthe peak of the wall shear-stress.

(a)

0 0.2 0.4 0.6 0.8 1-2

0

2

4

6

8

1010

-3

Reference

FF=2,W=4

(b)

0.2 0.25 0.35

5.5

6

6.5

7

7.510

-3

Reference

FF=2,W=4

Figure 9: The suction-side wall shear-stress comparison for the case FF2W4depicting (a)the chord-wise variation of τw and (b) the close-up region nearthe peak of the wall shear-stress.

to four chords downstream of the leading-edge (W = 4). The deviation of wallshear-stress nearly vanishes. Figure 9a shows the development of τw acrossthe airfoil which matches well with the reference case, and figure 9b shows theclose-up region near the peak τw where, unlike figure 8b, no large deviationfrom the reference case values is visible. Mean flow profiles (figure 10a) and

TBL around wing sections 121

100

101

102

103

0

5

10

15

20

25

Reference

FF=2,W=4

(a) Mean flow

100

101

102

103

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

(b) Reynolds stresses

Figure 10: Comparison of wall-normal profiles of (a) mean tangential-flowand (b) Reynolds stresses for the case FF2W4.

0 1 2 3 4-0.2

0

0.2

0.4

0.6

0.8

1

Reference

FF=2,W=4

Figure 11: Comparison of mean tangential-flow velocity profile in the sepa-rated region on the suction-side (x/c = 0.85).

Reynolds stress terms (10b), evaluated at x/c = 0.6 on the suction side, areexamined for this case. All the evaluated quantities show a good agreementwith the reference case values. Furthermore, mean velocity profile is evaluatedin the separated region at x/c = 0.85 and even in the separated region a verygood agreement is found as evident in figure 11. In the separated region τw isnearly zero, therefore the mean velocity is normalized using the far-field velocityof U∞ = 1.0. Figure 12a shows a 2D x-y section of the truncated domain withfigure 12b showing a close-up of the grid near the leading edge. The setup isvery similar to the Reference case (figure 6). Figure 13 shows the isocontours offlow structures identified by the λ2 criterion, which look qualitatively similar tothe ones seen in the reference case (figure 7).

A.3. Summary

Simulations of flow around a wing section with different domain sizes showthat for a C-grid type mesh topology, flow distortion effects due to domaintruncation and boundary conditions are minimized when the far-field boundaryis two chords away from the airfoil leading-edge and the outflow boundary isfour chords downstream of the leading-edge. The results are subject to theimposition of a RANS velocity field as Dirichlet boundary condition on the

122 Vinuesa et al.

(a) Simulation domain (b) Spectral-element grid

Figure 12: (a) Optimum truncated domain. (b) A close-up near the leadingedge of the orthogonal and structured spectral-element grid. Domain istruncated such that the far-field is 2 chords from the airfoil and the outflowis 4 chords downstream from the leading-edge.

Figure 13: Visualization of the instantaneous flow structures in the truncateddomain. The flow structures are identified by the isocontours of the λ2

criterion and are colored by streamwise velocity.

far-field boundaries and the energy-stabilized outflow boundary condition onthe outlet boundaries of the simulation.